§36.2 Catastrophes and Canonical Integrals

§36.2(i) Definitions

¶ Normal Forms Associated with Canonical Integrals: Cuspoid Catastrophe with Codimension

36.2.1

Special cases: , fold catastrophe; , cusp catastrophe; , swallowtail catastrophe.

¶ Normal Forms for Umbilic Catastrophes with Codimension

(elliptic umbilic).

(hyperbolic umbilic).

¶ Canonical Integrals

36.2.5.

with the contour passing to the lower right of .

with the contour passing to the upper right of .

¶ Diffraction Catastrophes

36.2.10.
36.2.11; .

For more extensive lists of normal forms of catastrophes (umbilic and beyond) involving two variables (“corank two”) see Arnol’d (1972, 1974, 1975).

§36.2(ii) Special Cases

36.2.12

is related to the Airy function (§9.2):

36.2.13

is the Pearcey integral (Pearcey (1946)):

(Other notations also appear in the literature.)

36.2.16

Addendum: For further special cases see §36.2(iv)

§36.2(iv) Addendum to 36.2(ii) Special Cases

36.2.28,
36.2.29.

Here the functions and are the complex conjugates of the functions and , respectively.